Question

Given three points (–1, 2), (0, 1), (1, –4) how do you write a quadratic function in standard form with the points?

Answer 1

`color(green)(y=-2x^2-3x+1)`

Explanation:

The generalized standard quadratic form is
`color(white)("XXX")ax^2+bx+c=y`

Substituting each of the given points: `(-1,2), (0,1), and (1,-4)`
gives three equations:
[1]`color(white)("XXX")a(-1)^2+b(-1)+c=2`
[2]`color(white)("XXX")a(0)^2+b(0)+c=1`
[3]`color(white)("XXX")a(1)^2+b(1)+c=0`

We can see directly from [2] that `c=1`

So [1] can be simplified as
[4]`color(white)("XXX")a-b+1=2`
and [3] can be simplified as
[5]`color(white)("XXX")a+b+1=-4`

Adding [4] and [5] gives
[6]`color(white)("XXX")2a+2=-2color(white)("XX")rarrcolor(white)("XX")a=-2`

Subtracting [4] from [5] gives
[7]`color(white)("XXX")2b=-6color(white)("XXXXX")rarrcolor(white)("XX")b=-3`

Substituting the calculated values for `a, b, and C` into the generalized quadratic form gives:
`color(white)("XXX")-2x^2-3x+1=y`